Tiling the grid with arbitrary tiles

3 March 2015
Vytautas Gruslys

Suppose that we have a tile $T$ in say $\mathbb{Z}^2$, meaning a finite subset of $\mathbb{Z}^2$. It may or may not be the case that $T$ tiles $\mathbb{Z}^2$, in the sense that $\mathbb{Z}^2$ can be partitioned into copies of $T$. But is there always some higher dimension $\mathbb{Z}^d$ that can be tiled with copies of $T$? We prove that this is the case: for any tile in $\mathbb{Z}^2$ (or in $\mathbb{Z}^n$, any $n$) there is a $d$ such that $\mathbb{Z}^d$ can be tiled with copies of it. This proves a conjecture of Chalcraft.

  • Combinatorial Theory Seminar